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We extend the classic convergence rate theory for subgradient methods toapply to non-Lipschitz functions. For the deterministic projected subgradientmethod, we present a global $O(1/\sqrt{T})$ convergence rate for any convexfunction which is locally Lipschitz around its minimizers. This approach isbased on Shor's classic subgradient analysis and implies generalizations of thestandard convergence rates for gradient descent on functions with Lipschitz orH\"older continuous gradients. Further, we show a $O(1/\sqrt{T})$ convergencerate for the stochastic projected subgradient method on convex functions withat most quadratic growth, which improves to $O(1/T)$ under either strongconvexity or a weaker quadratic lower bound condition.

Convergence Rates for Deterministic and Stochastic Subgradient Methods Without Lipschitz Continuity